Problem: A circle has a circumference of $12\pi$. It has an arc of length $\dfrac{15}{2}\pi$. What is the central angle of the arc, in degrees? ${12\pi}$ ${225^\circ}$ $\color{#DF0030}{\dfrac{15}{2}\pi}$
Solution: The ratio between the arc's central angle $\theta$ and $360^\circ$ is equal to the ratio between the arc length $s$ and the circle's circumference $c$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{s}{c}$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{15}{2}\pi \div 12\pi$ $\dfrac{\theta}{360 ^ \circ} = \dfrac{5}{8}$ $\theta = \dfrac{5}{8} \times 360 ^ \circ$ $\theta = 225^\circ$